Non-Markovian data-driven modeling of single-cell motility

Abstract

Trajectories of human breast cancer cells moving on one-dimensional circular tracks are modeled by the non-Markovian version of the Langevin equation that includes an arbitrary memory function. When averaged over cells, the velocity distribution exhibits spurious non-Gaussian behavior, while single cells are characterized by Gaussian velocity distributions. Accordingly, the data are described by a linear memory model which includes different random walk models that were previously used to account for various aspects of cell motility such as migratory persistence, non-Markovian effects, colored noise, and anomalous diffusion. The memory function is extracted from the trajectory data without restrictions or assumptions, thus making our approach truly data driven, and is used for unbiased single-cell comparison. The cell memory displays time-delayed single-exponential negative friction, which clearly distinguishes cell motion from the simple persistent random walk model and suggests a regulatory feedback mechanism that controls cell migration. Based on the extracted memory function we formulate a generalized exactly solvable cell migration model which indicates that negative friction generates cell persistence over long timescales. The nonequilibrium character of cell motion is investigated by mapping the non-Markovian Langevin equation with memory onto a Markovian model that involves a hidden degree of freedom and is equivalent to the underdamped active Ornstein-Uhlenbeck process.

Negative friction memory induces persistent motion

“…On the level of two-point correlation functions, such as the MSD, an (effective) equilibrium description may be obtained for a single degree of freedom by finding the corresponding kernel Γ (t) [26,44,46]. This has been of relevance for the interpretation of single particle tracking data of systems in and out of equilibrium where often only single, non-coupled degrees of freedom can be tracked [35,38,51].…”

“…Given the trajectory for a single Gaussian degree of freedom, unambiguous extraction of the underlying non-equilibrium dynamics is not possible, while the observed dynamics may be fully described by an effective equilibrium-like substitute model. Interestingly, these effective descriptions have led to memory kernels with negative tails [38], which might suggest the interpretation that energy is injected into the system at intermediate time scales. We anticipate that our results will be of further relevance for the analysis of single-particle tracking data of active systems.…”

“…For very pronounced negative tails and long memory times, i.e. bτ m −1 and τ τ m , we obtain the following scaling regimes in the MSD [38]:…”

“…For both the overdamped and underdamped cases, we show that these competing contributions to the memory kernel lead to a transient persistent regime in the MSD. Transient persistent motion appears in various models for self-propelled particles [17][18][19], transient dynamics of sheared systems [36,37] and singlecell motility [38]. Interestingly, the connection to negative memory contributions has been previously noted in the last two examples.…”

“…u (t) u (0) = v 2 0 e −|t|/τp , while the magnitude | u (t)| is not fixed. The MSD is then given by [38]…”